Question

Calculate the amount, in grams, of an original 300-gram sample of radioactive isotope Potassium-40 remaining after

3.9 billion years if its half life equals 1.3 x 10^9 years.

Answer 1

The current amount of the Potassium-40 sample is approximately 37.521 grams.

Step-by-step explanation:

The amount of the sample of the radioactive isotope decays exponentially in time, the amount of mass of the sample as a function of time (
`m (t)`), in grams, is described below:

`m(t) = m_(o)\cdot e^{-(t)/(\tau) }` (1)

Where:

`m_(o)` - Initial mass, in grams.

`t` - Time, in years.

`\tau` - Time constant, in years.

The time constant can be found from half life (
`t_(1/2)`), in years, described in statement:

`\tau = (t_(1/2))/(\ln 2)` (2)

If we know that
`m_(o) = 300\,g`,
`t = 3.9* 10^(9)\,yr` and
`t_(1/2) = 1.3* 10^(9)\,yr`, then the current amount of the sample is:

`\tau = (1.3* 10^(9)\,yr)/(\ln 2)`

`\tau \approx 1.876* 10^(9)\,yr`

`m = (300\,g)\cdot e^{-(3.9* 10^(9)\,yr)/(1.876* 10^(9)\,yr) }`

`m\approx 37.521\,g`

The current amount of the Potassium-40 sample is approximately 37.521 grams.